Magnetic Fields
from
Cosmological Perturbations
Keitaro Takahashi (Nagoya University, Japan)
1. Introduction
2. Electromagnetic properties of the early universe
3. Magnetogenesis & tight coupling approximation

1. Introductionprimordial fluctuations
observation
・CMB・galaxy distribution
theory・inflation(initial condition)
・cosmologicalperturbation theory(linear)magnetogenesis
magnetogenesis fromcosmological perturbations
before recombination
Hogan (2000)
Berezhiani & Dolgov (2004)
Matarrese et al. (2005)
Gopal & Sethi (2005)
KT et al. (2005, 2006, 2007, 2008)
Siegel & Fry (2006)Hollenstein et al. (2008)
Maeda et al. (2009)

based on・cosmological perturbation
theory (nonlinear)・observational facts
・no physical assumption

KT et al., Science 311 (2006) 827

basic idea
photons→ CMB

Thomsonscattering

baryonprotons
electrons
Coulomb
interaction

Thomson scattering→ deviation in motion
due to mass difference→ net electric charge densityand electric current→ magnetic fields

extensions to the conventional formalism
What do we need for magnetogenesis?
electric field and its rotationelectric field・Conventionally, baryons
・Separate treatment of p and e is necessary.
rotational part (Roy’s talk)・No rotational part at the linear order・generated by nonlinear effectLinear order is sufficient for CMB
but insufficient for B.
Two extensions are needed for magnetogenesis.

this talkunderstanding the physics of magnetogenesis
from cosmological perturbations●electromagnetic properties of the early universe
・solve Maxwell and Ohm
・Newtonian・neglect anisotropic stress● tight coupling approximation
・express B by familiar quantities (δγ)
・compare with another approach2. Electromagnetic propertiesof the early universe
KT, Ichiki & Sugiyama, PRD 77 (2008) 124028

EOMs(Newtonian) EOMs for photons, protons & electrons,
neglecting the anisotropic stress of photons.
ThomsonCoulomb

Thomsonsimplifying EOMs
relative and center-of-mass quantities

cosmological perturbations up to 2nd order→ Hall term and Lorentz force are higher order.

rewriting EOMs
p-e relative motion, γ, γ-baryon relative motion

generalized Ohm’s law

conventional CMB
Conventionally, we deal with photons and baryons.

Solving Maxwell + Ohm
basic equations
・EOM of photons
・EOM of relative motion
between photons and baryons
・generalized Ohm’s law
solve in this section
・Maxwell equations
how to solve
・up to 2nd order in cosmological perturbations
・regard Thomson term as an external source→ Electric charge, current and EM fields
are expressed as functions ofThomson term.

basic equations

EOM of Vpe↓
Ohm’s lawThomson term (source)

effective resistivity

first order in CP
B = 0 at the 1st order in CP.

First, take the divergence of the Ohm’s law.

charge density
divergence of the generalized Ohm’s law
damped oscillation
with a source

In cosmological timescale, plasma oscillation damps.
The equilibrium is nonzero due to the source.

solutions for the 1st order in CP

Electric charge,current and E field
are expressed by
the Thomson term.

second order in CP
B field joins at the second order.

solutions up to second order in CP

zero at the 1st orderinterpretation
photonE field pressure

protonelectron・electric current term is not important in Ohm’s law→ photon pressure balances with E field・current → (displacement current) + (B field)
・E and charge vanish when Thomson term disappear.
B and current do not because they are integral.

summary of section 2solving Maxwell + Ohm
・up to second order in CP・Thomson term was treated as an external source→ Electromagnetic quantities are expressedby the Thomson term.
・We need the Thomson term (photon-baryon
relative velocity) to evaluate B.

3. Magnetogenesis &tight coupling approximation

magnetogenesis
B field and Thomson term
vector product ofdensity gradient of photons
and velocity difference

vorticity
difference

We solve for δv by tight coupling approximation.
(Peebles & Yu, 1970; Kobayashi, Maartens, Shiromizu & KT, 2007)

EOM for γ-b relative motion

Electric current has a negligible effect on
the dynamics of the γ-b relative motion.

tight coupling approximation (TCA)Time derivativeis less important.

TCA I(I,1) → TCA I & CP 1, (I,2) → TCA I & CP 2

However, the source of B is zero at TCA I.

We must go on to TCA II.

TCA II

This is not parallel to ∇ρ in general.

B is generated at TCA II.

results

All EM quantities
are expressed byfamiliar quantities.

Evaluation of B spectrum is now in progress,
but the anisotropic stress must be included to be complete.

another approach
Ichiki, KT et al. (2006)
・derive the source term (relativistic)

・numerically calculate the spectrum
contributed from (1st order) × (1st order),neglecting the vorticity (purely 2nd order)

comoving B（log B (G))

spectrum
-20
-24
-28
-32
-36
-40
-44
1Gpc

1Mpc

scale

1kpc

1pc

toward the complete evaluation
numerical approach・numerically cancel TCA I
・include vorticity (purely 2nd order)
・confirm the validity of TCA
TCA approach
・TCA I already canceled by hand・include anisotropic stress
These two approaches are necessary
to cross check and evaluate the spectrum.

3 components
protonelectric current

electronbaryonphotontimescales

photon - baryon
center of mass
BecauseHτ << 1,we will use
tight coupling
approximation.

tight coupling approximation (TCA)
We need the velocity difference.

If (scattering time) << (dynamical timescale)tight coupling expansion is good. (Peebles & Yu, 1970)

TCA I
TCA IIEOM of photon-baryon relative motion

contribution from electric current

Electric current is negligible.
Protons and electrons can be treated as one fluid.

order estimation
deviation between photons and baryons

deviation between protons and electrons